Thinking
Mathematically
Combinations
A combination of items occurs when:
• The item are selected from the same
group.
• No item is used more than once.
• The order of the items makes no
difference.
Example Distinguishing between
Permutations and Combinations
For each of the following problems, explain if the
problem is one involving permutations or
combinations.
• Six students are running for student government
president, vice-president, and treasurer. The
student with the greatest number of votes becomes
the president, the second biggest vote-getter
becomes vice-president, and the student who gets
the third largest number of votes will be student
government treasurer. How many different
outcomes are possible for these three positions?
Solution
• Students are choosing three student
government officers from six candidates.
The order in which the officers are chosen
makes a difference because each of the
offices (president, vice-president, treasurer)
is different. Order matters. This is a
problem involving permutations.
Example Distinguishing between
Permutations and Combinations
For each of the following problems, explain if the
problem is one involving permutations or
combinations.
• Six people are on the volunteer board of
supervisors for your neighborhood park. A threeperson committee is needed to study the
possibility of expanding the park. How many
different committees could be formed from the six
people on the board of supervisors?
Solution
• A three-person committee is to be formed
from the six-person board of supervisors.
The order in which the three people are
selected does not matter because they are
not filling different roles on the committee.
Because order makes no difference, this is a
problem involving combinations.
Example Distinguishing between
Permutations and Combinations
For each of the following problems, explain if
the problem is one involving permutations
or combinations.
• Baskin-Robbins offers 31 different flavors
of ice cream. One of their items is a bowl
consisting of three scoops of ice cream,
each a different flavor. How many such
bowls are possible?
Solution
• A three-scoop bowl of three different flavors is to
be formed from Baskin-Robbin’s 31 flavors. The
order in which the three scoops of ice cream are
put into the bowl is irrelevant. A bowl with
chocolate, vanilla, and strawberry is exactly the
same as a bowl with vanilla, strawberry, and
chocolate. Different orderings do not change
things, and so this problem is combinations.
Combinations of n Things Taken r at a
Time
The number of combinations possible if r
items are taken from n items is
n!
n Cr 
(n  r)!r!
Example Using the Formula for
Combinations
A three-person committee is needed to study
ways of improving public transportation.
How many committees could be formed
from the eight people on the board of
supervisors?
Solution
The order in which the three people are
selected does not matter. This is a problem
of selecting r = 3 people from a group of n =
8 people. We are looking for the number of
combinations of eight things taken three at a
time.
8!
8! 8 7  6  5!


 56
8 C3 
(8  3)!3! 5!3!
5!3  2  1
Example Using the Formula for
Combinations
In poker, a person is dealt 5 cards from a
standard 52-card deck. The order in which
you are dealt the 5 cards does not matter.
How many different 5-card poker hands are
possible?
Solution
Because the order in which the 5 cards are
dealt does not matter, this is a problem
involving combinations. We are looking for
the number of combinations of n=52 cards
drawn r=5 a a time.
52!
52!

52 C5 
(52  5)!5! 47!5!
52  51 50  49  48 47!

 2,598,960
47!5 4  3  2 1
Example Using the Formula for
Combinations and the Fundamental
Counting Principle
The U.S. Senate of the 104th Congress
consisted of 54 Republicans and 46
Democrats. How many committees can be
formed if each committee must have 3
Republicans and 2 Democrats?
Solution
The order in which members are selected does
not matter. We begin with the number of
ways of selecting 3 Republicans out of 54
Republicans without regard to order.
54!
54!

54 C3 
(54  3)!3! 51!3!
54  53 52  51!

 24,804
51!3 2 1
Solution cont.
Next, we find the number of ways of selecting
2 Democrats out of 46 Democrats without
regard to order.
46!
46!
46  45 44!


 1035
46 C2 
(46  2)!2! 44!2!
44!2 1
Solution cont.
We use the Fundamental Counting Principle
to find the number of committees that can
be formed:
28,804  1035 = 25,672,140
Thus, 25,672,140 committees can be formed.
Thinking
Mathematically
Combinations